1. Options and Speculative Markets 2005-2006 Greeks
Professor André Farber
Solvay Business School
Université Libre de Bruxelles

2. Fundamental determinants of option value
Call value
Put Value
Current asset price S
Delta 
0 < Delta < 1 
- 1 < Delta < 0
Striking price K
Interest rate r
Rho
Dividend yield q
Time-to-maturity T
Theta ?
Volatility
Vega
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3. Example
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4. Delta
Sensitivity of derivative value to changes in price of underlying asset
Delta = ∂f / ∂S
As a first approximation : f ~ Delta x S
In example, for call option : f = Delta = 0.637
If S = +1: f = → f ~
If S = 101: f = error because of convexity
Binomial model: Delta = (fu – fd) / (uS – dS)
European options: Delta call = e-qT N(d1) Delta put = Delta call - 1
Forward : Delta = + 1
Call : 0 < Delta < +1
Put : -1 < Delta < 0
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5. Calculation of delta
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6. Variation of delta with the stock price for a call
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7. Delta and maturity
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8. Delta hedging
Suppose that you have sold 1 call option (you are short 1 call)
How many shares should you buy to hedge you position?
The value of your portfolio is:
V = n S – C
If the stock price changes, the value of your portfolio will also change.
V = n S - C
You want to compensate any change in the value of the shorted option by a equal change in the value of your stocks.
For “small” S : C = Delta S
V = 0 ↔ n = Delta
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9. Effectiveness of Delta hedging
This diagram illustrate the effectiveness of delta hedging.
The initial stock price is S = 100
You are are short on 1 call option. The value of this option is
To hedge you position, you buy Delta = shares.
Suppose that the stock price suddenly drops to 95.
The value of your short call option drops to Since you are short, you gain ( – 7.511) = on your short call position.
On the other hand, you are long on shares.
As the price change is S = -5, you loose (-5)(0.637) = on you share.
The net result in a change in the value of you portfolio:
V = =
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10. Gamma = ∂Delta / ∂S = ∂²f / ∂S²
A measure of convexity
Gamma = ∂Delta / ∂S = ∂²f / ∂S²
Taylor: df = f’S dS + ½ f”SS dS²
Translated into derivative language:
f = Delta S + ½ Gamma S²
In example, for call : f = Delta = Gamma = 0.019
If S = +1: f = ½ → f ~
If S = 101: f =
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11. Variation of Gamma with the stock price
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12. Gamma and maturity
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13. Gamma hedging Back to previous example.
We have a delta neutral portfolio:
Short 1 call option
Long Delta = shares
The Gamma of this portfolio is equal to the gamma of the call option:
V = n S – C →∂V²/∂S² = - Gammacall
To make the position gamma neutral we have to include a traded option with a positive gamma. To keep delta neutrality we have to solve simultaneously 2 equations:
Delta neutrality
Gamma neutrality
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14. Theta Measure time evolution of asset Theta = - ∂f / ∂T
(the minus sign means maturity decreases with the passage of time)
In example, Theta of call option =
Expressed per day: Theta = / 365 = (in example)
Theta = / 252 = (as in Hull)
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15. Variation of Theta with the stock price
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16. Relation between delta, gamma, theta
Remember PDE:
Theta
Delta
Gamma
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